Let `U_(1) and U_(2)` be two urns such that `U_(1)` contains 3 white and 2 red balls, and `U_(2)` contains only 1 white ball. A fair coin is tossed. If head appears, than 1 ball is drawn at random for `Us_(1)` and put into `U_(2),` However, if tail appears, then 2 balls are drawn at random from `U_(1)` and put into `U_(2).` Now 1 ball is drawn at random from `U_(2)`
Given that the drawn ball from `U_(2)` is white, the probability that head appeared on the coin is

Options

A. `(17)/(23)`
B. `(11)/(23)`
C. `(15)/(23)`
D. `(12)/(23)`

Answer

D

Explanation

`P("Head appeared/white from" U_(2))`
`=P(H)*({(""^(3)C_(1))/(""^(5)C_(1))xx(""^(2)C_(1))/(""^(2)C_(1))+(""^(2)C_(1))/(""^(5)C_(1))xx(""^(1)C_(1))/(""^(2)C_(1))})/(23//30)`
`=(1)/(2)({(3)/(5)+1+(2)/(5)xx(1)/(2)})/(20//30)`
`(12)/(23)`